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Course Queries Syllabus Queries 2 years ago
Posted on 16 Aug 2022, this text provides information on Syllabus Queries related to Course Queries. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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In the syllabus for my course on group theory, there is the following theorem which is proven: "If H⊂ZnH⊂Zn is a subgroup, then H≅ZkH≅Zk for 0≤k≤n0≤k≤n".
I understand most of the proof, but there's a simple line that is intuitive, yet doesn't mathematically make sense to me.
In the proof, which is done by induction, the following homomorphism is defined
The kernel of the map is considered, which are all vectors of the form (m1,...,mn,0)(m1,...,mn,0).
Then the syllabus reads "Since H⊂Zn+1H⊂Zn+1 is a subgroup, also H∩ Ker REPLY 0 views 0 likes 0 shares Facebook Twitter Linked In WhatsApp
You actually already said the answer. Ker(π)Ker(π) is isomorphic to ZnZn and H∩Ker(π)H∩Ker(π) is a subgroup of Ker(π)Ker(π), and so we can consider it as a subgrouop of ZnZn.
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